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I stumbled onto a question in my studies today that I am not sure how to resolve and you seem like just the person to ask. The question is this: what, exactly, makes a paradox different from a regular old argument? Consider: we tend to call paradoxes those arguments which seem sound and yet whose conclusions we are not inclined to accept. Hence, what one of my professors calls Meno’s Paradox is not a paradox in Meno’s eyes. For him it’s simply an argument that shows we can’t come to know things. I think the same can be said for Zeno’s paradoxes. Zeno was not trying to conclude with contradictions for us to be puzzled over—he was trying to give reductio ad absurdum arguments against motion and time. If Zeno was right about time and motion then none of his arguments are paradoxes any more than the problem of evil is a paradox for the atheist. It seems to me that the only thing that makes a paradox a paradox is that the consumer is unwilling to accept its conclusion (or has independent reason to think the conclusion must be wrong). Am I missing something here?
What is the difference between a paradox and an argument? An excellent question the answer to which depends on how 'paradox' and 'argument' are defined. Following Nicholas Rescher, I would define a paradox as a set of individually plausible but collectively inconsistent propositions. Meno's paradox, also known as the paradox of inquiry, is an example. It can be cast in the form of the following aporetic tetrad:
1. Either we know what we are seeking or we don't.
2. If we know what we are seeking, then inquiry is unnecessary.
3. If we don't know what we are seeking, then inquiry is impossible.
4. It is not the case that inquiry is either unnecessary or impossible.
Each of these propositions is plausible. And yet they cannot all be true. Any three of them entails the negation of the fourth. Thus, from (1), (2), and (3) we get the negation of (4). And from (2), (3), and (4), we derive the negation of (1). And so on.
The above aporetic tetrad is not an argument, but a mere set of inconsistent propositions. There are, however, four distinct arguments that can be generated from it. What then is the difference between a paradox and an argument? A paradox or aporetic polyad (as I would prefer to call it) is a set of individually plausible but jointly inconsistent propositions. An argument is a set of propositions together with the claim that one of them, the conclusion, follows from the others, the premises.
Thus Meno in the eponymous Platonic dialog is not presenting a problem or paradox for our consideration, but arguing for a thesis, namely, that inquiry is either unnecessary or impossible. Thus he is arguing from (1), (2), and (3) to the negation of (4). Socrates/Plato, on the other hand, is arguing from (2), (3), and (4) to the negation of (1). The doctrine of anamnesis is in effect the negation of (1): it is not the case that we must either explicitly know what we are seeking or explicitly not know what we are seeking. We can have a sort of implicit, tacit, pre-thematic, unarticulated knowledge. Inquiry, accordingly, is the making explicit of this implicit knowledge.
To sum up, paradoxes (aporetic polyads) and arguments are closely related but distinct. They are closely related because for every n-membered polyad, there are n corresponding arguments, and for every n-premised argument there is a corresponding polyad with n + 1 members. They are distinct because aporetic polyads are not intended to establish one of their members as true on the basis of the others taken as given. But in an argument, the whole point is to establish as true one proposition on the basis of the others taken as granted.
Example. Corresponding to the hoary syllogism 'All men are mortal; Socrates is a man; ergo, Socrates is mortal" there is the corresponding antilogism the limbs of which are the premises of the syllogism plus the negation of its conclusion. And corresponding to this antilogism are three syllogisms: the original one and "Socrates is a man; Socrates is not mortal; ergo, not all men are mortal" and " Socrates is mortal; Socrates is not a man; ergo, not all mortals are men."
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